5. ORIGIN OF THE
M -
RELATION

The greatest dividend to come so far from the
M -
relation
has been the resolution of the apparent discrepancy between black hole
masses in nearby galaxies, the masses of black holes in AGN,
and the mass density in black holes needed to explain quasar light.
But the importance of the
M -
relation presumably goes
beyond its ability to clarify the data.
Like other tight, empirical correlations in astronomy, the
M -
relation must be telling us
something fundamental about
origins, and in particular, about the connection between black
hole mass and bulge properties.

Probably the simplest way to relate black holes to bulges
is to assume a fixed ratio of
M to
Mbulge. Since
M (Eq. 2),
this assumption implies
Mbulge.
In fact this is well known to be the case: bulge luminosities scale as
~ 4, the
Faber-Jackson law, and mass-to-light ratios scale as ~
L1/4
(Faber et al. 1987),
giving Mbulge ~
5, in agreement
with the slope = 4.5±0.5
derived above for the
M -
relation.

On the other hand, the
M -
relation appears to be much
tighter than
the relation between and
bulge mass or luminosity.
And even if a tight correlation between black hole mass and bulge
mass were set up in the early universe, it is hard to see how
it could survive mergers, which readily convert disks to bulges
and may also channel gas into the nucleus, producing
(presumably) uncorrelated changes in
M and
Mbulge.
The tightness of the
M -
relation suggests that some
additional
feedback mechanism acts to more directly connect black hole masses
to stellar velocity dispersions and to maintain that connection
in spite of mergers.

One such feedback mechanism was suggested by
Silk & Rees (1998)
even before the discovery of the
M -
relation.
These authors explored a model in which supermassive black holes
first form via collapse of ~ 106M gas clouds
before most of the bulge mass has turned into stars.
The black holes created in this way would then accrete and radiate,
driving a wind which acts back on the accretion flow.
Ignoring star formation, departures from spherical symmetry etc.,
the flow would stall if the rate of deposition of mechanical energy
into the infalling gas was large enough to unbind the protogalaxy
in a crossing time TD.
Taking for the energy deposition rate some fraction f of the
Eddington luminosity LE, we have

(3)

Writing
GMbulge2Rbulge,
TDRbulge /
and LE = 4cGM /
with
the opacity,

(4)

consistent with the observed relation.
The constant of proportionality works out to be roughly correct if
f ~ 0.01 - 0.1
(Silk & Rees 1998).

This model assumes that black holes
acquire most of their mass during a fast accretion phase,
tacc
107 yr.
Kauffmann & Haehnelt
(2000)
developed a semi-analytic model
for galaxy formation in which black holes grow progressively
larger during galaxy mergers.
The cooling of the gas that falls in during mergers
is assumed to be partially balanced by energy input from supernovae.
This feedback is stronger for smaller galaxies which has
the effect of steepening the resulting relation between
M
and .
Haehnelt & Kauffmann
(2000)
found M ~
3.5 but
the slope could easily have been increased if the feedback
had been set higher (M. Haehnelt, private communication).
However the scatter in the
M -
relation derived by them
was only slightly less than the scatter in
M vs
Lbulge,
in apparent contradiction with the observations
(Figure 1).

Burkert & Silk (2001)
also considered a model in which black
holes grow by accreting gas during mergers.
In their model, accretion is halted when star formation begins to
exhaust the gas in the outer accreting disk;
the viscous accretion rate is proportional
to, and assuming a star formation time scale that is
proportional to TD, Burkert & Silk found
MRbulge2/GMbulge,
with a constant
of proportionality that is again similar to that observed.
This model does not give a convincing explanation for the tight
correlation of
M with
however.

Feedback of a very different sort was proposed by
Norman, Sellwood &
Hasan (1996),
Merritt & Quinlan
(1998)
and Sellwood & Moore
(1999).
These authors simulated the growth of massive compact objects at
the centers of barred or triaxial systems and noted how the
nonaxisymmetric component was weakened or dissolved when the
central mass exceeded a few percent of the stellar mass.
Since departures from axisymmetry are believed to be crucial for
channeling gas into the nucleus, the growth of the black hole
has the effect of cutting off its own supply of fuel.
These models, being based purely on stellar dynamics, have the
nice feature that they can be falsified, and in fact they
probably have been:
our new understanding of black hole demographics
(Section 2.3) suggests
that few if any galaxies have
M /
Mbulge as great as 10-2.
(At the time of these studies,
several galaxies were believed from ground-based data
to have
M /
Mbulge > 1%, including NGC 1399
(Magorrian et al. 1998),
NGC 3115
(Kormendy et al. 1996a),
and NGC 4486b
(Kormendy et al. 1997)).

The tightness of the
M -
relation must place strong
constraints on the growth of black holes during mergers.
We know empirically that mergers manage to keep galaxies on
the fundamental plane, which is a relation between
,
the bulge effective radius Re and the surface brightness
at Re.
The that appears in the
fundamental plane relation
is the same c
that appears in the
M -
relation (indeed,
it was defined by Ferrarese & Merritt 2000 for just this reason)
and furthermore c
is defined within a large enough aperture that
it is unlikely to be significantly affected by dynamical processes
associated with the formation of a black-hole binary during a merger
(Milosavljevic &
Merritt 2001).
Hence the physics of the black-hole binary can be ignored and
we can ask simply: How do mergers manage
to grow black holes in such a way that
logM
4.5
log, independent of changes
in Re and L?

This work was supported by NSF grant 00-71099 and
by NASA grants NAG5-6037 and NAG5-9046.
We thank B. Peterson and A. Wandel for useful discussions.